The remainder when 10¹⁰ + 10¹⁰⁰ + 10¹⁰⁰⁰ + . . . . . . + 10^1000000000 is divided by 7 is

Explanation:-

Number of terms in the series = 10.
(We can get it easily by pointing the number of zeros in power of terms.
In 1st term number of zero is 1, 2nd term 2, and 3rd term 3 and so on.)
$$\eqalign{ & \frac{{{{10}^{10}}}}{7},\,{\text{Written}}\,{\text{as}}, \cr & \frac{{{{\left( {7 + 3} \right)}^{\left( {4 \times 2 + 2} \right)}}}}{7} \cr & {\text{The}}\,{\text{remainder}}\,{\text{will}}\,{\text{depend}}\,{\text{on}}, \cr & \frac{{{3^2}}}{7} \cr & {\text{So,}}\,{\text{remainder}}\,{\text{will}}\,{\text{be}}\,2 \cr & \frac{{{{10}^{1000}}}}{7},\,{\text{remainder}} = 2 \cr & \frac{{{{10}^{10000}}}}{7},\,{\text{remainder}} = 1 \cr & {\text{So, we get alternate 2 and 1 as remainder, five times each}}{\text{.}} \cr & {\text{So, required remainder is given by}} \cr & \frac{{\left( {2 + 1 + 2 + 1 + 2 + 1 + 2 + 1 + 2 + 1} \right)}}{7} \cr & = \frac{{15}}{7} \cr}$$